{
  "id": "2303.04853",
  "version": "v2",
  "published": "2023-03-08T19:40:10.000Z",
  "updated": "2024-08-31T02:13:29.000Z",
  "title": "A Host--Kra ${\\mathbf F}_2^ω$-system of order $5$ that is not Abramov of order $5$, and non-measurability of the inverse theorem for the $U^6({\\mathbf F}_2^n)$ norm",
  "authors": [
    "Asgar Jamneshan",
    "Or Shalom",
    "Terence Tao"
  ],
  "comment": "70 pages, no figures. Minor changes in notation (particularly with regards to the concept of a k-cocycle) and some new remarks",
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "It was conjectured by Bergelson, Tao, and Ziegler that every Host--Kra ${\\mathbf F}_p^\\omega$-system of order $k$ is an Abramov system of order $k$. This conjecture has been verified for $k \\leq p+1$. In this paper we show that the conjecture fails when $k=5, p=2$. We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $f: {\\mathbf F}_2^n \\to {\\mathbf C}$ of large Gowers norm $\\|f\\|_{U^6({\\mathbf F}_2^n)}$ which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial $e(P)$, but with the property that all such phase polynomials $e(P)$ are ``non-measurable'' in the sense that they cannot be well approximated by functions of a bounded number of random translates of $f$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-08-31T02:13:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28D15",
      "37A05",
      "37A35"
    ],
    "keywords": [
      "inverse theorem",
      "non-measurability",
      "non-classical quintic phase polynomial",
      "large gowers norm",
      "conjecture fails"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}